sequential probability ratio test

The Neyman-Pearson lemma says that the likelihood ratio test is the uniformly most powerful test at a given sample size. But the likelihood ratio is a nonnegative supermartingale, hence an e-process, so it provides type-I error control at all times, and is a sequential test. How does it perform as such?

Let $(X_t)$ be a stream of data and let $L_t={\prod }_{i\le t}q(X_1)/p(X_1)$ be the likelihood ratio at time $t$ for a simple alternative $Q$ with density $q$ against a simple null $P$ with density $p$. Suppose you have desired type-I error of $\alpha$ (false positives) and desired type-II error of $\beta$ (false negatives). Then there exist thresholds $t_1(\alpha ,\beta )$ and $t_2(\alpha ,\beta )$ such that if you reject the null when $L_t\ge t_1(\alpha ,\beta )$ and reject the alternative when $L_t\ge t_2(\alpha ,\beta )$ such that the test is optimal. Here optimality means that, among all tests with the same (or lower) type-I and type-II error rates, the SPRT requires the fewest number of samples in expectation.

Wald introduced this test in the 1930s, and it was proved optimal by Wald and Wolfowitz in 1948.

Unfortunately, $t_1$ and $t_2$ are often impossible to solve for, so approximations are necessary. One usually sets $t_1=(1-\beta )/\alpha$ and $t_2=\beta /(1-\alpha )$. Note that if we set $\beta =0$, meaning we want a power 1 test (rejects the null with probability 1 in the limit under the alternative), then we recover the threshold $\log (1/\alpha )$ for rejecting the null, which is what one would usually obtain for sequential testing with Ville’s inequality.

Reading

• Optimum Character of the Sequential Probability Ratio Test.
• Improving approximate SPRT: https://arxiv.org/pdf/2410.16076. When the distributions are discrete, the thresholds $t_1(\alpha )$ and $t_2(\beta )$ may not be precisely attainable, in which case the test will be overly conservative.